# Week 8: General Form, Quadratics

This week I learned many cool and fascinating facts about the general form of a quadratic equation ( $ax^2 + bx +c$ ) as well as analyzing $y=a (x-p)^2 + q$

General form

$x^2$ :

• Positive – Open upward
• Negative – Opens downwards

bx:

• is a mixture of both $ax^2$ and bx so it can differ

c:

• Is the y axis

Analyzing equation:  $y=a (x-p)^2 + q$

using this formula can tell us a lot of things that will help us graph the equation. In fact it will tell us 8 helpful clues.

ex. \$latex -2(x+2)^2 – 1

1. Vertex: the vertex will be p and q … (-2,-1) * p is always backwards from the formula because in the original equation it is (x – p) therefore it has to be a negative number for it to become positive in the new equation.
2. Axis of symmetry: This will be p because it is the x value on a graph …-2
3. Opens up or down: This is determined if $x^2$ is negative or positive. If it is negative it will be opening down, and if it is positive it will be opening upwards… Opens down because it is -2
4. If it is congruent to $y = x^2$. This means that the pattern of the graph would be 1, 3, 5. Meaning 1 over 1 up, 1 over 3 up, 1 over 5 up etc. If it does not follow the 1,3,5 rule it means that a has a value different from -1 or 1. Say a = 4, multiply 4 to each number from the original pattern. ex/ 1,3,5 is now equal to 4, 12 , 20 etc. This pattern will also tell us if the parabola will be stretched or compressed. If 1> a < 0 ( fraction ) it means that the parabola will be compressed. If  1 < a > 1 it means that the parabola will be stretched…. this equation is congruent to y = $-2x^2$, it is being stretched because 2 > 1, it is negative so it will be flipped upside down but the structure is the same. (congruent)
5. Minimum or maximum: If a is positive, it will have a minimum. If a is negative it will have a maximum… It has a maximum of -1
6. Domain. We know the XER because the parabola never ends.
7. Range: If a is positive y will be greater than the minimum. If a is negative, y will be smaller than the maximum… y < -1, yER

IMPORTANT

\$latex y = 3x^2 (x – 3)^2 + 4

The horizontal translation will be 3 units to the right even though it says -3.

I also learned that general form can be turned into the analyzing equation of  $y=a (x-p)^2 + q$ by completing the square.

Here is a video that helped me understand how to complete the square

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